Properties

Label 576.4.a.k
Level $576$
Weight $4$
Character orbit 576.a
Self dual yes
Analytic conductor $33.985$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 576.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(33.9851001633\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 8)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2 q^{5} + 24 q^{7} + O(q^{10}) \) \( q - 2 q^{5} + 24 q^{7} - 44 q^{11} - 22 q^{13} - 50 q^{17} - 44 q^{19} + 56 q^{23} - 121 q^{25} + 198 q^{29} - 160 q^{31} - 48 q^{35} + 162 q^{37} + 198 q^{41} - 52 q^{43} - 528 q^{47} + 233 q^{49} - 242 q^{53} + 88 q^{55} - 668 q^{59} - 550 q^{61} + 44 q^{65} - 188 q^{67} - 728 q^{71} + 154 q^{73} - 1056 q^{77} - 656 q^{79} + 236 q^{83} + 100 q^{85} - 714 q^{89} - 528 q^{91} + 88 q^{95} - 478 q^{97} + O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 0 0 −2.00000 0 24.0000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.4.a.k 1
3.b odd 2 1 64.4.a.d 1
4.b odd 2 1 576.4.a.j 1
8.b even 2 1 72.4.a.c 1
8.d odd 2 1 144.4.a.e 1
12.b even 2 1 64.4.a.b 1
15.d odd 2 1 1600.4.a.o 1
24.f even 2 1 16.4.a.a 1
24.h odd 2 1 8.4.a.a 1
40.f even 2 1 1800.4.a.d 1
40.i odd 4 2 1800.4.f.u 2
48.i odd 4 2 256.4.b.a 2
48.k even 4 2 256.4.b.g 2
60.h even 2 1 1600.4.a.bm 1
72.j odd 6 2 648.4.i.h 2
72.n even 6 2 648.4.i.e 2
120.i odd 2 1 200.4.a.g 1
120.m even 2 1 400.4.a.g 1
120.q odd 4 2 400.4.c.i 2
120.w even 4 2 200.4.c.e 2
168.e odd 2 1 784.4.a.e 1
168.i even 2 1 392.4.a.e 1
168.s odd 6 2 392.4.i.g 2
168.ba even 6 2 392.4.i.b 2
264.m even 2 1 968.4.a.a 1
264.p odd 2 1 1936.4.a.l 1
312.b odd 2 1 1352.4.a.a 1
408.b odd 2 1 2312.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.4.a.a 1 24.h odd 2 1
16.4.a.a 1 24.f even 2 1
64.4.a.b 1 12.b even 2 1
64.4.a.d 1 3.b odd 2 1
72.4.a.c 1 8.b even 2 1
144.4.a.e 1 8.d odd 2 1
200.4.a.g 1 120.i odd 2 1
200.4.c.e 2 120.w even 4 2
256.4.b.a 2 48.i odd 4 2
256.4.b.g 2 48.k even 4 2
392.4.a.e 1 168.i even 2 1
392.4.i.b 2 168.ba even 6 2
392.4.i.g 2 168.s odd 6 2
400.4.a.g 1 120.m even 2 1
400.4.c.i 2 120.q odd 4 2
576.4.a.j 1 4.b odd 2 1
576.4.a.k 1 1.a even 1 1 trivial
648.4.i.e 2 72.n even 6 2
648.4.i.h 2 72.j odd 6 2
784.4.a.e 1 168.e odd 2 1
968.4.a.a 1 264.m even 2 1
1352.4.a.a 1 312.b odd 2 1
1600.4.a.o 1 15.d odd 2 1
1600.4.a.bm 1 60.h even 2 1
1800.4.a.d 1 40.f even 2 1
1800.4.f.u 2 40.i odd 4 2
1936.4.a.l 1 264.p odd 2 1
2312.4.a.a 1 408.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(576))\):

\( T_{5} + 2 \)
\( T_{7} - 24 \)
\( T_{11} + 44 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( 2 + T \)
$7$ \( -24 + T \)
$11$ \( 44 + T \)
$13$ \( 22 + T \)
$17$ \( 50 + T \)
$19$ \( 44 + T \)
$23$ \( -56 + T \)
$29$ \( -198 + T \)
$31$ \( 160 + T \)
$37$ \( -162 + T \)
$41$ \( -198 + T \)
$43$ \( 52 + T \)
$47$ \( 528 + T \)
$53$ \( 242 + T \)
$59$ \( 668 + T \)
$61$ \( 550 + T \)
$67$ \( 188 + T \)
$71$ \( 728 + T \)
$73$ \( -154 + T \)
$79$ \( 656 + T \)
$83$ \( -236 + T \)
$89$ \( 714 + T \)
$97$ \( 478 + T \)
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